Consider the following recurrence relation: $$\begin{array}{rl}{a}_0& =\frac{\sqrt{5}+1}{2}\\ a_{n+1}& ={\displaystyle \frac{a_n(a_n^4+10a_n^2+5)}{5a_n^4+10a_n^2+1}}\end{array}$$

Note that $a_0$ is the golden ratio.

$a_n$ can always be written in the form ${\displaystyle \frac{p_n\sqrt{5}+1}{q_n}}$, where $p_n$ and $q_n$ are positive integers.

Let $s(n)=p_n^5+q_n^5$. So, $s(0)=1^5+2^5=33$.

The Fibonacci sequence is defined as: $F_1=1$, $F_2=1$, $F_n=F_{n-1}+F_{n-2}$ for $n>2$.

Define ${\displaystyle S(m)=\sum _{i=2}^{m}s(F_i)}$.

Find $S(1618034)$. Submit your answer modulo $398874989$.